How many critical points are there in $(\,x- 1\,)(\,x- 2\,)\,...\,(\,x- 2020\,)$?

Question

How many critical points are there in $$(\,x- 1\,)(\,x- 2\,)\,...\,(\,x- 2020\,)$$ My observation is

There are $7$ critical points (${\rm C.P}$) in $$(\,x- 1\,)(\,x- 2\,)\,...\,(\,x- 8\,)$$ $[\,{\rm C.P},\,x\,]\in (\,1,\,2\,),\,(\,2,\,3\,),\,...\,,(\,7,\,8\,)$

There are $8$ critical points (${\rm C.P}$) in $$(\,x- 1\,)(\,x- 2\,)\,...\,(\,x- 9\,)$$ $[\,{\rm C.P},\,x\,]\in (\,1,\,2\,),\,(\,2,\,3\,),\,...\,,(\,7,\,8\,),\,(\,8,\,9\,)$

...

There are $2019$ critical points (${\rm C.P}$) in $$(\,x- 1\,)(\,x- 2\,)\,...\,(\,x- 2020\,)$$ $[\,{\rm C.P},\,x\,]\in (\,1,\,2\,),\,(\,2,\,3\,),\,...\,,(\,2017,\,2018\,),\,(\,2018,\,2019\,),\,(\,2019,\,2020\,)$

But I can find the way to use it there, I need to the help! Thanks for all the nice comments and your interests! (I also guess by using discriminant, so I think this may help us!) Good luck everybody! $29$.

Answer 1

You're almost done! Note that the function is differentiable everywhere, and that there are $2020$ zeroes. Thus, by Rolle's Theorem, there are at least $2019$ critical points. Since the derivative of the function is a polynomial of degree $2019,$ then it has at most $2019$ zeroes by the Fundamental Theorem of Algebra, meaning that your function has at most $2019$ critical points.

Answer 2

Apply Rolle theorem for each of the intervals you mentioned. You cannot have more than 2020-1=2019 critical points because your derivative is a polynomial of a degree 2020-1=2019